Derivation of the exponential distribution and Poisson distribution and binomial distribution using stochastic differential equations

The exponential distribution and Poisson distribution have traditionally been explained as follows: “The exponential distribution is the distribution of the time interval between the occurrence of rare events, and the Poisson distribution is the distribution of the number of occurrences of rare events”. In addition, the Poisson distribution has been derived from the binomial distribution using a method based on the Poisson's law of small numbers. However, from a logical consideration of depression, it was found that it is not possible to say that “the distribution of the time interval between occurrences of rare events is an exponential distribution, and the distribution of the number of occurrences of rare events is a Poisson distribution” unless there is an approximation that the time during which events that rarely occur are occurring is negligible. Therefore, we considered the exponential distribution and Poisson distribution while taking into account the time during which events are occurring. As a result, it was found that the distribution of the total time, including the time when the event is occurring and the time when it is not occurring, is an exponential distribution, and the distribution of the number of times the event occurs is a Poisson distribution. Next, we were able to give a stochastic differential equation from which the exponential distribution and Poisson distribution can be derived. In Addition, it was found that the Poisson distribution can be derived without using the assumption of the Poisson's law of small numbers, as in the past, and that the distribution of the number of times any event occurs follows the Poisson distribution. In addition, it was also found that the results obtained in this study encompass the conventional interpretation if the time during which events are occurring is very short. We also gave the stochastic differential equation that gives the binomial distribution.